Study of (1) Everolimus, (2) Estrogen Deprivation Therapy (EDT) With Leuprolide + Letrozole and (3) Everolimus + EDT in Patients With Unresectable Fibrolamellar Hepatocellular Carcinoma (FLL-HCC)

Inclusion Criteria: * Patients ≥ 12 years old. * Pathologically confirmed diagnosis of advanced and/or unresectable FLL-HCC. This will be performed by the participating centers on submitted specimens. If the submitted material is insufficient for analysis, a repeat biopsy is recommended. * ECOG performance status 0-2 ; Lansky performance score of ≥ 60% for patients 12-16 years old * Adequate hematologic, renal and hepatic function defined as: Hematologic: ANC ≥ 1.0 x 10\^9/L, platelets ≥ 50 x 10\^9/L o Renal: creatinine ≤ 2 x upper limit of normal, or creatinine Clearance of ≥60 cc/mL/1.73 m\^2 for patients \> 16 years old. For patients ≤ 16 years of age, creatinine Clearance of ≥70 cc/mL/1.73 m\^2 or serum creatinine based on the following chart: Creatinine clearance or radioisotope GFR ≥ 70ml/min/1.73 m\^2 or serum creatinine based on age/gender as follows: Age Maximum Serum Creatinine (mg/dL) Male Female 10 to \< 13 years 1.2 1.2 13 to \< 16 years 1.5 1.4 ≥ 16 years 1.7 1.4 The threshold creatinine values in this Table were derived from the Schwartz formula for estimating GFR (Schwartz et al. J. Peds, 106:522, 1985) utilizing child length and stature data published by the CDC. * Hepatic: total bilirubin ≤ 2 mg/dL, alanine and aminotransferase levels ≤ 5 x upper limit of normal for age. * Fasting blood glucose \<1.5 x upper limit of normal . If fasting glucose \> 1.5 x upper limit of normal, adequate glycemic control (fasting glucose \< 1.5 x upper limit of normal ) for three weeks is recommended before starting protocol therapy. * At least 1 target lesion measurable by Response Evaluation Criteria in Solid Tumors (RECIST 1.1) guidelines. * Target lesion(s) must not lie within a previously resected, irradiated, ablated, or chemoembolized area. If a lesion does lie in such an area, there must be evidence of a ≥ 20% increase in diameter and/or the appearance of a new lesion on subsequent imaging in order for such a lesion to be considered a target lesion. * Prior systemic therapy is allowed. Prior surgery, locoregional ablative or embolic therapies are also permitted provided that the criteria for measurable disease as outlined above are met. * Concurrent antiviral therapy for hepatitis B is permitted * Women of childbearing potential must be practicing an effective method of birth control that may include intrauterine devices (both hormonal and non-hormonal are acceptable), double-barrier method, male partner sterilization or abstinence, before enrollment, and throughout the study and for 6 months after receiving the last dose of study drug. * Men must agree to use a double barrier method of birth control and to not donate sperm during the study and for 6 months after receiving the last dose of study drugs. Sperm banking is acceptable for interested male patients enrolled on study prior to initiating treatment. Prescription oral contraceptives, contraceptive injections, and contraceptive patch are not approved methods of contraception in this study. * Negative pregnancy test (serum hCG) result (applicable to women of child bearing potential) within 7 days before Cycle 1 Day 1 of study treatment. Exclusion Criteria: * Concurrent anticancer, or radiation therapy. Patients must have completed all anticancer therapy \> 4 weeks before the start of study therapy. The date of last palliative radiation must be \> 2 weeks from the start of study therapy. Palliative radiation is permitted on protocol with MSK PI discretion on treatment modifications. * Patients, who have had a major surgery or significant traumatic injury within 4 weeks of start of study drug, patients who have not recovered from the side effects of any major surgery (defined as requiring general anesthesia) . * Patients receiving chronic, systemic treatment with corticosteroids or another immunosuppressive agent. Topical or inhaled corticosteroids are allowed. * Concurrent oral contraceptive use or hormonal replacement therapy. * Use of an aromatase inhibitor, GnRH agonist and/or tamoxifen within the past 30 days. Patients previously on fulvestrant or a q3 month GnRH agonist must have discontinued these medications for at least 3 months. * Concurrent use of potent CYP3A4 and/or P-glycoprotein inhibitors or potent CYP3A4 inducers (please see Appendices 3 and 4). Where possible, otherwise eligible patients should be switched to alternative agents; otherwise, they will be excluded from the study. * Potent CYP3A4 inducers decrease serum everolimus levels and should not be given concomitantly. Dose modifications of everolimus are not indicated in the presence of moderate CYP3A4 inducers \[108\]. Please refer to Appendix 3 for a complete list of potent and moderate inducers of CYP3A4. * Potent CYP3A4 and/or P-glycoprotein inhibitors can increase serum levels of everolimus and should not be co-administered. Moderate inhibitors may mildly-moderately increase serum everolimus levels, though there is no definitive evidence supporting a dose reduction \[108\]. Please refer to Appendix 4 for a complete list of potent and moderate inhibitors of CYP3A4. * Any investigational drug received within one month of study enrollment. * Any severe, uncontrolled medical conditions that, in the opinion of the investigator, may be exacerbated by study therapy including infection, diabetes and cardiopulmonary disease. * Any psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant or nursing women. * Patients with a known hypersensitivity to everolimus, letrozole, leuprolide and/or related compounds or their excipients. * Patients who received any form of transplant and who are on any form of immunosuppressive therapy. However transplanted patients who are off immunosuppressive therapy for at least 4 weeks are allowed on the study, provided that any of their immunosuppressive-related toxicities have recovered to at least a grade 1. * Known HIV positive with a CD4 count \< 500 cells/mm3. * Immunization with a live vaccine \< 1 week of initiating study therapy or during therapy. * BSA \<1 m\^2